Non-Lipschitzian Control Algorithm for Nanoscale
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Non-Lipschitzian Control Algorithm for Nanoscale Friction V. Protopopescu, J. Barhen, and Y. Braiman Center for Engineering Science Advanced Research Computing and Computational Sciences Directorate Oak Ridge National Laboratory, Oak Ridge, TN 37831
We present a robust feedback control algorithm and apply it to the nonlinear oscillator array (Frenkel-Kontorova) model of nanoscale friction. The new control approach is based on the concepts of non-Lipschitzian dynamics and global targeting. We show that average quantities of the controlled system can be driven – exactly or approximately – towards desired targets which become additional, linearly stable attractors for the system’s dynamics. Extensive numerical simulations show that the basins of attraction of these targets are reached in very short times and the control exhibits very strong robustness. We investigate the efficiency of the control in terms of various parameters (e.g., system size, non-Lipschitzian exponent).
1. INTRODUCTION The ability to control and manipulate friction during sliding is extremely important for a large variety of technological applications. The outstanding difficulties in realizing efficient friction control are related to the complexity of the task, namely dealing with nonlinear systems consisting of many degrees of freedom, under strict size confinement, and with only very limited control access. Nonlinear systems – especially when driven far from equilibrium – may exhibit a wide variety of spatial and temporal behaviors, each resulting in different patterns of motion, ranging from stable and unstable periodic behavior to fully chaotic regimes. Maintaining the stability of the existing regime and/or changing the type of dynamics are often needed in practical applications and are most often realized by using external controls. Traditionally, friction has been manipulated by applying small perturbations to accessible elements and parameters of the sliding system. This operation requires a-priori knowledge of the strength and timing of the perturbations. Recently, the groups of J. Israelachvili [1] (experimental) and U. Landman [2] (full-scale molecular dynamics computer simulation) showed that friction in thin- film boundary lubricated junctions can be reduced by coupling the small amplitude (of the order of 1Å) directional mechanical oscillations of the confining boundaries to the molecular degree of freedom of the sheared interfacial lubricating fluid. Significant changes in frictional responses were observed also in the two-plate model [3] by modulating the normal response to lateral motion [4]. In addition, surface roughness and thermal noise are thought to play a
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significant role in deciding upon control strategies at the micro and the nano-scale [5,6]. All these methods rely essentially on non- feedback control. In this paper, we present a recently proposed non-Lipschitzian feedback control scheme [7, 8] as applied to the nonlinear Frenkel-Kontorova model nanoscale friction [9]. In general, feedback methods are
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